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Aronszajn, Krzywicki and Szarski proved in \cite{AKS62} a strong uniquecontinuation result for differential forms, satisfying a certain first orderdifferential inequality, on Riemannian manifolds with empty boundary. Thepresent paper extends this result to the setting of Riemannian manifold withnon-empty boundary, assuming suitable boundary conditions on the differentialforms. We then present some applications of this extended result. Namely, weshow that the Hausdorff dimension of the zero set of harmonic Neumann andDirichlet forms, as well as eigenfields of the curl operator (on$3$-manifolds), has codimension at least $2$. Again, these bounds were known inthe setting of manifolds without boundary, so that the merit is once more theinclusion of boundary points.

A unique continuation theorem for exterior differential forms on Riemannian manifolds with boundary